CHAPTER 4 THE TIME VALUE OF MONEY

Chapter outline Introduction Interest rates Future value and compounding of lump sums Compounding interest more frequently than annually Nominal and effective interest rates Present value and discounting More on present and future values Valuing annuities Perpetuities Amortising a loan Sinking funds Conclusion

Learning outcomes By the end of this chapter, you should be able to: use various computation tools to analyse the role of time value in finance calculate, interpret and explain the future value and present value of single amounts or lump sums, and investigate the relationship between them calculate, interpret and explain the future value and present value of annuities (ordinary annuities, annuities due and ordinary deferred annuities) calculate, interpret and explain the present value of a perpetuity

Learning outcomes (cont.) By the end of this chapter, you should be able to: calculate and interpret the present value and future value of a mixed stream of cash flows determine deposits needed to accumulate a future sum, calculate installments to amortise a loan and calculate an interest rate or growth rate calculate the present value, future value, interest rate and time period using discounting and compounding principles.

Introduction The value of an investment depends on: size timing of cash flows The larger the cash inflows, and the sooner the receipt of these cash flows, the more valuable the investment Time value of money: Cash flows to be received in the near future are more valuable than ones to be received in the distant future

Interest rates Simple interest Compound interest Interest earned on the principal amount only – interest earned is not reinvested Compound interest All interest earned is reinvested together with principal amount – interest is earned on original principal as well as on interest that has been reinvested

Example 4.1 Simple interest Sibusiso receives a R1 000 bonus. He invests the R1 000 in a savings account that offers a simple interest rate of 10% p.a. for a period of five years. How much money will Sibusiso have after five years? Initial principal Interest Year 1: 10% of R1 000 = R100 Year 2: 10% of R1 000 = R100 Year 3: 10% of R1 000 = R100 Year 4: 10% of R1 000 = R100 Year 5: 10% of R1 000 = R100   Initial principal: R1 000 Total interest earned over this period: R 500 Final amount after five years: R1 500

Example 4.1 Compound interest Sibusiso invests R1 000 in a savings account offering interest at 10% p.a. but the interest earned will be re-invested. How much money will Sibusiso have after five years? Initial Previous Principal Interest Principal New amount Year 1: 10% of R1 000,00 = R100,00 R1 000,00 R1 100,00 Year 2: 10% of R1 100,00 = R110,00 R1 100,00 R1 210,00 Year 3: 10% of R1 210,00 = R121,00 R1 210,00 R1 331,00 Year 4: 10% of R1 331,00 = R133,10 R1 331,00 R1 464,10 Year 5: 10% of R1 464,10 = R146,41 R1 464,10 R1 610,51   Initial principal: R1 000,00 Final amount after five years: R1 610,51 Total interest earned over this period: R 610,51

Future value and compounding of a lump sum Future value (FV): determine accumulated value of all cash flows at end of a project (Tn) Present value (PV): discounts all cash flows to start/beginning of a project (time zero, T0) FVn = PV0 × (1+i)n

Example 4.2 Sibusiso invests his money for a period of one year at an interest rate of 10%. What will the FV of his investment be at the end of the year?

Example 4.3 Sibusiso invests his money for two years at 10%. Calculate the FV of his investment after two years.

Compounding interest more frequently than annually Interest often computed more frequently than once a year Terminology for different frequencies of compounding: Nominal annual rate compounding annually (NACA) Nominal annual rate compounding semi-annually (NACSA) Nominal annual rate compounding quarterly (NACQ) Nominal annual rate compounding monthly (NACM)

Semi-annual, quarterly and monthly compounding Formula when interest is compounded more than once per period: FVn = PV0 × FV increases when frequency of compounding the interest payments is increased

Example 4.6 Sibusiso invests his money for a period of two years at an interest rate of 10%, compounded semi-annually. What will the FV of his investment be at the end of this period?

Continuous compounding Interest sometimes computed continuously FVn = PV0 × e i × n

Nominal and effective interest rates Nominal (stated) interest rate: contractual annual percentage rate of interest charged by a lender or promised by a borrower Effective (true) annual rate: annual rate of interest actually paid or earned. Effective annual rate includes effects of compounding frequency; nominal rate does not EAR =

Example 4.10 What is the effective annual rate of interest if an annual nominal rate of 8% is compounded quarterly?   Using the formula: EAR = = = (1,02)4 − 1 = 0,0824 = 8,24%

Present value and discounting Present value (PV) Amount of money invested today at given interest rate for specified period to equal future amount Alternatively: PV is amount today that is equivalent to future payment that has been discounted by appropriate interest rate Since money has time value: PV of future amount is worth less longer you have to wait to receive it Process of finding PVs: discounting

Present value and discounting PV Amount of money that would have to be invested today at given interest rate over specified period to equal future amount PV0 =

Example 4.11 Fikile wishes to find the current value (PV) of an amount of R1 700 that will be received eight years from now, assuming that the annual interest rate is 8%.   Using the formula: PV = FV × (1 + i)-n = R1 700 × (1,08)-8 = R1 700 × 0,5402 = R918,46

More on present and future values Determining an interest rate Sometimes necessary to calculate the return on investment Calculating the number of periods Sometimes necessary to work out the number of time periods for an initial investment to accumulate to a given FV

Valuing annuities Annuity Two types of annuities Series of equal payments (cash outflows) or receipts (cash inflows) occurring over specified time period Consists of constant payments made at regular intervals (monthly, quarterly, annually, etc.) Two types of annuities Ordinary annuity (or annuity in arrears): payments or receipts occur at the end of each period Annuity due (or annuity in advance): payments occur at the start of each period

Example 4.14 You deposit R2 000 at the end of each of the next five years in an account that pays 10% interest p.a. What will the FV of your account be after five years?

Example 4.14 FVA = PMT ×

FV of an annuity due Annuity due: each payment occurring is moved ahead one period in order to convert the cash flow stream into an ordinary annuity Achieved by multiplying PMT by (1+i) Resulting cash flows of R5 300 (5 000 × 1,06) at the end of each period are similar to the cash flows of R5 000 at the beginning of each period

Example 4.19 What amount will accumulate if you deposit R5 000 at the beginning of each year for the next five years in an account with an interest of 6% compounded annually?

PV of an ordinary annuity PVA: current value of a stream of expected or promised future payments that have been discounted to a single equivalent value today PVA = PMT ×

Example 4.20 Suppose you need an investment that will pay R2 000 at the end of every year for the next five years at an annual interest rate of 10%. How much should you invest today?

Example 4.22 What amount must you invest today at 6% interest compounded annually so that you can withdraw R5 000 at the beginning of each year for the next five years?

Mixed stream of cash flows Annuity based on equal payments over a number of periods During capital budgeting cash flows from initial investment usually not in form of annuity Unequal cash flows will probably be generated over project lifetime In some cases, positive as well as negative cash flows may occur Mixed cash flow stream Not possible to use annuity formulae and calculator solutions

Example 4.24

Example 4.24

Example 4.24

Perpetuities Perpetuity: annuity in which periodic payments begin on a fixed date and continue indefinitely (perpetual annuity) Three types of perpetuities: Ordinary perpetuity: payments made at the end of the stated periods Perpetuity due: payments made at the beginning of the stated periods Growing perpetuity: periodic payments grow at a given rate (g)

Perpetuities Formula used to calculate the PV of an ordinary perpetuity (PV∞):   PV∞ =    Formula used to calculate the PV of a growing perpetuity:

Conclusion A lump sum refers to a single payment or receipt of cash at a specific point in time. A distinction was made between initial cash flows (occurring at time zero, i.e. now) and future cash flows that occur somewhere in future. An annuity can be defined as a stream of equal, periodic cash flows over a specified period of time, in equally spaced time intervals. These payments are usually annual, but can occur at other intervals, such as monthly (e.g. bond payments). Annuity formulae allow complex problems to be resolved in a systematic manner.

Conclusion (cont.) A perpetuity is a perpetual stream of constant or constantly growing cash flows. A mixed cash stream consists of non-constant cash flows, where different cash flows occur every period. The FVs and PVs of lump sum amounts, annuities and mixed cash flows can be calculated by making use of formulae, financial tables or a financial calculator.

Conclusion (cont.) Loan amortisation refers to the determination of equal loan payments (the extinction of a debt by means of equal periodic payments over a period of time). Therefore, amortisation is a schedule showing the repayment details for a loan, including the amount of each payment that is apportioned to interest and to capital (the principal debt). Sinking funds are used to accumulate money over time, by depositing periodic payments in a fund. Examples of sinking funds are to make provision for the replacement of assets, or to make provision for a loan that needs to be repaid.